Motivation

• Expert decision-makers make consequential decisions based on predictions of an unknown outcome
• Common Situation in Economics:
  • Judges deciding pretrial release.
    • Judges must predict whether a defendant will fail to appear in court
  • Doctors making diagnoses
    • A prediction if a patient suffers some undiagnosed condition
  • Managers making hiring decisions
    • Which applicants should be hired based on predictions of their future productivity

Empirical Example

For a particular judge, we can observe their release rate at each predicted risk decile. We can also observe the failure to appear rate. Question: Are these patterns of pretrial release decisions rationalisable as maximising expected utility at accurate predictions of failure to appear risk given some private information and a utility function that can vary across the four cells?

Key Questions

Do experts make systematic prediction mistakes based on the available information?

If so: - Which decision makers? - On which decisions? - In which ways are predictions systemically biased?

Observable Data

• We observe dataset \((X, C, Y) \sim_{i.i.d} \mathcal{P}\) of each decision made by a DM. For each decision we observe
  • Characteristics \(X\)
  • DM’s Choice \(C\)
  • Selectively observed outcome \(Y = C \cdot Y^*\)

Identification

• Analyse DM’s choices through a nonparametric expected utility maximisation problem, given:
  • Subjective beliefs about outcome given characteristics (Unknown)
  • Utility function (Unknown)
  • Private information (Unknown)
• Given all the unknown factors can we answer the question: Are DM’s subjective beliefs about the outcome given characteristics accurate? i.e. Lie in the identified set for \(P(Y^* |X)\)
• If not, conclude MS’s choices reflect systematic prediction mistakes.

Identification

• Analyse DM’s choices through a nonparametric expected utility maximisation problem, given:
  • Subjective beliefs about outcome given characteristics (Unknown)
  • Utility function (Unknown)
  • Private information (Unknown)
• How do we draw inferences about the DM’s subject beliefs about \(Y^* \mid X\) using \((X, C, Y) \sim \mathcal{P}\) without knowledge of DM’s Utility function and private information.

Results

1. Identification: Are DM’s choices consistent with expected utility at accurate beliefs
   • Econometric assumptions addressing missing data problem, and behaviour assumptions on the DMs utility function

Results

1. Identification: Are DM’s choices consistent with expected utility at accurate beliefs
   • Econometric assumptions addressing missing data problem, and behaviour assumptions on the DMs utility function
2. Characterisation: In what ways are the DM’s predictions systematically biased?
   • Bound how common and costly are systematic prediction mistakes
   • Bound whether DM’s subjective beliefs overreact or under-react to characteristics
   • Quasi-experimental variation \(+\) out-of-sample prediction \(\to\) confidence intervals

Information and Beliefs

Information and Beliefs

- No assumptions are placed on the distribution of \(V\) and we will model non-parametrically

Information and Beliefs

Information and Beliefs

\[ \underbrace{Q(Y^* \mid V, X)}_{\text{Posterior}} \propto \underbrace{Q(V \mid Y^*, X)}_{\text{Likelihood}} \underbrace{Q(Y^* \mid X)}_{\text{Subjective Beliefs}} \] - Goal: Draw Inferences about DM’s subject beliefs \(Q(Y^* \mid X)\) using \((X, C, Y) \sim \mathcal{P}\), using data on their choices which reflect their posterior, leaving the private information and likelihood unspecified.

Utility function

• Suppose research partitions \(X = (X_{0}, X_{1})\)

• \(u(c, y^*; x_{0})\): Payoff of choice \(c\) at outcome $y^* and characteristics \(x_{0}\)

• Exclusion Restriction: Characteristics \(X_{0}\) directly effect both utility function and beliefs, whereas other characterises \(X_{1}\) and private information \(V\) only affect beliefs.

Utility function

• Suppose research partitions \(X = (X_{0}, X_{1})\)

• \(u(c, y^*; x_{0})\): Payoff of choice \(c\) at outcome $y^* and characteristics \(x_{0}\)

• Exclusion Restriction: Characteristics \(X_{0}\) directly effect both utility function and beliefs, whereas other characterises \(X_{1}\) and private information \(V\) only affect beliefs.

\(\implies\) Conditional on \(X_{0}\), variation in DM’s choice probabilities across \(X_{1}\) only reflect variation in posterior beliefs \(Q(Y^* \mid V, X)\) but not variation in utility function \(u(c, y^* ; x_{0})\)

Utility function

• Suppose research partitions \(X = (X_{0}, X_{1})\)

• \(u(c, y^*; x_{0})\): Payoff of choice \(c\) at outcome $y^* and characteristics \(x_{0}\)

• Exclusion Restriction: Characteristics \(X_{0}\) directly effect both utility function and beliefs, whereas other characterises \(X_{1}\) and private information \(V\) only affect beliefs. ### Example

• Pretrial Release: Judges utility function only directly depends on

  • Defendant’s race (Discrimination)
  • Defendant’s age
  • Defendant’s charge severity

Utility Maximisation

• Given utility function, expected utility maximisation yields set of feasible joint distributions that must satisfy the following two outcomes: $$ (X, V, C, Y^*) Q \begin{cases}

1. C {c’} {Q}[u(c’, Y^*; X_{0}) | X, V], \
2. C Y^* X, V. \end{cases} $$

• We can say that a DM’s observed choices are consistent with expected utility maximisation, at accurate beliefs, if there exists a utility function and associated joint distribution \(Q\) under expected utility model s.t.: \[ \underbrace{Q(X, C, Y)}_{\text{model}} = \underbrace{P(X, C, Y)}_{{\text{data}}}, \text{ where } Y = C \cdot Y^* \] We are able to construct a utility function, accurate beliefs, private information, tie-breaking rule that would reproduce observable data (the decision maker’s choices).

Interpretation

• The DM is making a Systematic Prediction Mistake based on \(X\) if choices are inconsistent with expected utility maximisation at accurate beliefs
• There is no combination of utility function, accurate beliefs, private information, tie-breaking rule that would reproduce observable data (the decision maker’s choices).

Pretrial Release

• Focus on a class linear utility functions. (Linear in outcome)
  • Costly to release defendants that would fail to appear and detail defendants that would not fail to appear
• Are DM’s choices consistent w/ expected utility maximisation at accurate beliefs, any private information, and any linear utility function?

Characterisation

To show a DMs choices are consistent w/ expected utility max. at any accurate beliefs, linear utility, private information, it is sufficient to show that \(\forall x_{0}\) \[ \max_{\tilde{x}_{1}} \mathbb{E}\left[ \sum_{k} Y_{k}^* \mid C=1, X=(x_{0}, \tilde{x}_{1}) \right] \leq \min_{\tilde{x}_{1}} \mathbb{E}\left[ \sum_{k} Y_{k}^* \mid C=0, X=(x_{0}, \tilde{x}_{1}) \right] \] given \[ \underbrace{Y^* \mid \{ C=1, X \}}_{\text{point identified}} \text{ and } \underbrace{Y^* \mid \{ C=0, X \}}_{\text{not identified}} \] - We observe conditional distribution of \(Y^*\) for those released. We do not for those detained.

\(\implies\) DM’s choices are inconsistent w/ expected utility max. at accurate beliefs iff there exists no \(Y^* \mid \{ C=0 , X \}\) that satisfies the inequalities.

Characterisation

\[ \max_{\tilde{x}_{1}} \mathbb{E}\left[ \sum_{k} Y_{k}^* \mid C=1, X=(x_{0}, \tilde{x}_{1}) \right] \leq \min_{\tilde{x}_{1}} \mathbb{E}\left[ \sum_{k} Y_{k}^* \mid C=0, X=(x_{0}, \tilde{x}_{1}) \right] \] given \[ \underbrace{Y^* \mid \{ C=1, X \}}_{\text{point identified}} \text{ and } \underbrace{Y^* \mid \{ C=0, X \}}_{\text{not identified}} \]

• Intuitively, over the class of linear utility functions, EU max. requires the DM to make choices according to an incomplete threshold rule on their posterior beliefs. This is showing that expected utility max. at any accurate beliefs and private information is observationally equivalent to behaviour generated by EU max. under beliefs governed by these two inframarginal conditional expectations.
• We only need to check if we could reproduce the decision makers choices under a threshold rule on these inframaringal conditional expectations, one of which is identified, the other which is not identified, but for which we are able to get informative bounds using quasi-experimental variation.

Problem

With the assumptions so far, the DM’s choices are always consistent w/ EU max. at some linear utility function and accurate beliefs.

break

We therefore, need to make the following assumption

Econometric Assumptions to construct informative bounds on missing data

\[ \underline{\mathbb{E}} \left[ \sum_{k}Y_{k}^* \mid C=0, X \right] \leq {\mathbb{E}} \left[ \sum_{k}Y_{k}^* \mid C=0, X \right] \leq \overline{\mathbb{E}} \left[ \sum_{k}Y_{k}^* \mid C=0, X \right] \leq \]

Behavioural Assumptions that exclude characteristics \(X_{1}\) and private information \(V\) from directly affecting the utility function.

Problem

With the assumptions so far, the DM’s choices are always consistent w/ EU max. at some linear utility function and accurate beliefs.

break

We therefore, need to make the following assumption:

1. Econometric Assumptions to construct informative bounds on missing data

\[ \underline{\mathbb{E}} \left[ \sum_{k}Y_{k}^* \mid C=0, X \right] \leq {\mathbb{E}} \left[ \sum_{k}Y_{k}^* \mid C=0, X \right] \leq \overline{\mathbb{E}} \left[ \sum_{k}Y_{k}^* \mid C=0, X \right] \]

2. Behavioural Assumptions that exclude characteristics \(X_{1}\) and private information \(V\) from directly affecting the utility function.

We estimate these bounds through

Instrumental Variable Bounds

Observed FTA rates

Question: Are these patterns of pretrial release decisions rationalisable as maximising expected utility at accurate predictions of failure to appear risk given some private information and a utility function that can vary across the four cells?

Bounds

And to do that, I use the quasi-random assignment of judges to cases and apply those instrumental variable bounds that I didn’t have the chance to talk through in more details to construct this curve in blue, which is an upper bound on the failure to appear rate among defendants detained by this particular judge at each defendant cell and each predicted risk decile. Now the identification results simply ask us whether there are any misrankings in this judge’s choices. And it turns out based on the point estimate, the answer is yes. This judge is releasing defendants at the top of the predicted risk distribution at a higher observed failure to appear risk while simultaneously detaining defendants at the bottom of the predicted risk distribution that have a strictly lower worst case failure to appear risk. They could exchange their choices and do strictly better no matter their utility function and private information.

These are point estimates; really, there should be standard errors around them. So formally, we would test

Implementation

For each judge: 1. Calculate release rates by defendant cells 2. Observe failure rates for released defendants 3. Bound failure rates for detained defendants 4. Test for choice misrankings 5. Account for statistical uncertainty

Key Findings

• At least 20% of judges make systematic mistakes
• Mistakes concentrated in:
  • Tails of risk distribution
  • Cases with black defendants
• Results robust to various specifications

Conclusions

• Framework enables testing for prediction mistakes under weak assumptions
• Can characterize magnitude and nature of mistakes
• Results informative for algorithm adoption decisions
• Extensions to learning, dynamics remain open questions

Thank You